Directional computer



May 9, 1967 G. ARSHAL 3,319,052

DIRECTIONAL COMPUTER Original Filed July 2, 1962 7 Sheets-Sheet 1REFERENCE VOLTAGE MOTORTACH INTEGRATOR TACH FEEDBACK MOTOR-TACHINTEGRATOR TACH FEEDBACK MOTOR-TACH INTEGRATOR TACH FEEDBACK INVENTOR.

y 9 967 G. ARSHAL 3,319,052

DIRECTIONAL COMPUTER Original Filed July 2, 1962 '7 Sheets-Sheet L "IGYRO REFERENCE FRAME (Y2 41 6) FIG. 10

IN V EN TOR.

G. ARSHAL May 9, 1967 DIRECTIONAL COMPUTER 7 Sheets Sheet 25 OriginalFiled July 2 1962 n Nal wsamu N al INVENTOR.

May 9, 1967 Original Filed July 2.

G. ARSHAL DIRECTIONAL COMPUTER 7 Sheets-Sheet 4 AMP FIG. 1c

I I I I I I 8 o 5 i I ll ll I ll N E 7 r. OLE

r: O. l O. 0. CL 2 2 2 2 2 1 4 4 q 4 A A 1 AH N I) -o -1,

INVENTOR.

M y 9. 1967 G. ARSHAL 3,319,052

DIRECTIONAL COMPUTER Original Filed July 2, 1962 '7 Sheets-Sheet E FRAME"2 l w b,

CROSS PRODUCT b COMPUTATION o (41 be a x F COMPUTATION F l G. 1dINVENTOR.

y 9, 1967 G. ARSHAL 3,319,052

DIRECTIONAL COMPUTER Original Filed July :3, 1962 '7 Sheets-Sheet 6 FIG.2

I FLIGHT CONTROL SYSTEM R T F HT TR A C VELOCI' I DI EC IONAL LlG CON ULA'RCRAFT COMPUTER L COMPUTER FIG. 3

IN V EN TOR.

y 9, 1967 G. ARSHAL 3,319,052

DIRECTIONAL COMPUTER Original Filed July 2, 1962 '7 Sheets-Sheet r FIG.4 FIG. 5

g g cos 1; g sin 17 /g g: TRANSFORMATION c 9 FROM 92 g2 sin 17 93 cos 1o H VERTICAL GYRO T0 FREE emu OF FIG 2 I I?) turf (g /g MOTOR APPLY AS 0m FIG. 2 l -g,,

FIG. 6

I N VEN TOR.

United States Pat'ent 3,319,052 DIRECTIONAL COMPUTER George Arshal, 714Ardmore Ave., Redlands, Calif. 92373 Continuation of application Ser.No. 206,801, July 2, 1962. This application Jan. 20, 1966, Ser. No.533,106 25 Claims. (Cl. 235-15026) This application is a continuation ofpending patent application No. 206,801 of filing date July 2, 1962, nowabandoned.

The invention is a vectorial data processing system for extracting orprescribing directions and their rates of change from given inputvectors. It offers efliciency and versatility for producing thedirection in satisfaction of any and all geometrical and rate of changeproperties that are possibly possessed by a direction. While offeringthis facility, the directional computer can be realized in conjunctionwith any physically demonstrated frame of reference.

The directional computer proceeds on the principle that a directionchanges when it is given an angular velocity. Any input vector can enterthe directional computer and assert the angular velocity of the outputdirection in one or both of two ways:

(1) So that the output direction rotates around the input vector todescribe a conical surface.

(2) So that the output direction rotates directly towards the inputvector and seeks coalignment.

The rate of action in each case depends on the magnitude of the inputvector. The rates are independently controllable by modifying the inputvector through dis tinct scalar coefiicients. This holds implicationsbeyond a complete capability of controlling the directional rates ofchange. The scalar coefficient controlling either mode of directionalrate of change can clearly be made zero to stOp this process, or it canbe made negative to reverse the process. Where it is desired to rotatethe output direction around the input vector (mode 1 above) to aparticular azimuth, it is only necessary to make the scalar coefiicientcontrolling this rate of turn responsive to the deviation of the outputdirection from the desired azimuth. In like manner, the scalarcoefficient controlling the output directions rate of turn towards theinput vector (mode 2 above) can guide this process into a desiredrelative angle. The scalar coefficient simply has to register thedeviation of the output direction from its desired angle relative to theinput vector.

These capabilities of the directional computer support applicationsexceeding the stabilization functions of one kind or another associatedwith previous directional output devices. They additionally servicedemanding applications requiring wide ranging directional controls undera variety of options.

Aircraft flight control offers a case in point. Very effective flightcontrol can be achieved by automatically controlling the aircraft'svelocity to a given direction. Thereafter, the direction is solelyresponsible for guiding the aircraft. The direction must be producedwith the versatility of generating any flight path desired.

FIGURE 1 and any one of FIGURES la, lb, 1c and ld together draw anembodiment of the directional computer. FIGURE 2 is another embodiment.FIGURE 3 illustrates an application of the directional computer. FIGURES4 and 5 are geometrical representations of basic processes supported bythe computer. FIGURE 6 illustrates a specifically identified inputvector under the embodiment of FIGURE 2.

The directional computer, in all its embodiments, is an implementationof the relationship:

Hie- MW where 3 represents the directional output vector; Y representsany given vector or the result of a combination of vectors; A and Brepresent scalar constants. Under this formulation, 5 is maintained at aconstant length, having therefore a purely directional significance, andits direction is controlled with complete versatility by the choice ofY.

These events are understood from the derivative form of (1):

d? A B)b+F Y 2) If (2) is multiplied out scalarly by 5-, therelationship establishing its magnitude is isolated as This is a linear,first order differential equation in 11 Under this relationship, b soonassumes the value A/B and then holds it indefinitely. Thereafter,relation (2) is represented in its essential characteristics by:

since this relation does not disturb the magnitude of '5. If (4) ismultiplied vectorially by its action is described in full by:

The left side of (5) is identically equal to the angular velocity of band the right side of (5) produces it as the portion of Y that isperpendicular to 5.

Since the input vector Y is open to choice, it can be stated as:

where Z is a given vector. Then relation (1) becomes:

and has the alternative form:

This follows from the continued vector product formula. The coefficientb multiplying Z in (8) can be replaced by the value A/B given to it byrelation (3). That is, the input (6) can be asserted in relation (1) as:

ess of extracting its direction is a variable computational convenience.Also, the rate of change d5 d dt di i immediately available as 5x (2x?)itself.

Possible vector inputs as Y or Z are represented in such measurements asvelocity, acceleration, range, hearing, navigational heading andvertical reference outputs, and the like. The data are intelligible onlyin reference to particular defined directions where the directions aredefined by being exhibited. As such, the measurements express vectors.They can be utilized to express Y or 2 as desired. In general, a vectorconsists in one or more scalar values assigned to one or more physicallydefined directions. This is all thats necessary to express Y or 7Specific examples of Y or Z are readily selected to suit specificapplications of the directional computer. The problem of aircraft flightcontrol offers a useful demonstration of this.

An aircraft maneuvers about by the act of turning its velocity andaccepts guidance through its ability to control the direction of itsvelocity. As a first requirement of developing such guidance, thedirection of flight must be prescribed and given to the aircrafts flightcontrol system. The output 5 of the directional computer provides thatservice. In addition, the rate of change of 5 must be supplied to theflight control system. The aircraft can sustain its velocity along agiven, changing direction only if it develops the necessaryaccelerations. d'B/dt is essential for an assertion of theseacceleration requirements. It is immediately available as the integrandEXT controlling the direction of F. Finally, the direction 5 must beresponsive to navigational data, which may be entered manually as pilotcommanded signals, or automatically by various navigational instruments,or by a combination of manual and automatic means. The inputs Y and 2given to the directional computer can accommodate any such requirements.

FIGURE 3 is a block diagram of this flight control scheme. The flightcontrol computer operates the aircrafts control surfaces to maintaincoalignment between the direction 5 and the aircraft velocity 7.Subsequently, the direction '6 flies the aircraft. Such automaticcontrols are an essential part of any automatic aircraft guidance systemand serve further, under manual guidance, to overcome deficiencies thatmight otherwise be present in the handling characteristics of theaircraft. The inputs Y and 2 shown entering the directional computer areexamples composed by the quantities 5, C, 9, and E. The vector 5represents the local vertical direction that is furnished aboard theaircraft by a vertical gyro; the scalars C and 9 represent quantitiesthat are expressly produced to prescribe a desired climb angle and adesired horizontal rate of turn in 5. Under this development, C and Q inFIGURE 3 can be generated manually in lieu of manually applied elevatorand aileron deflections.

The input 2 in relation (7) can be qualified so that the quantity Ceffects any desired climb angle in the output 5. In function, relation(7) is the same as and it drives 77 into the direction of Z. This actionis immediately evident from the geometrical representation of FIGURE 4.For interpretive convenience, 5 may be regarded here as having unitlength. Then the quantity 5x (2X73) is just the vector component of Ethat is perpendicular to F. F proceeds under such rate of change intothe direction of E and then persists along 2. The convergence is fast orslow depending on the magnitude of Z.

If 2 is expressed as:

where a is a scalar coefficient and a 0, U is driven into the directionof Z"; that is: the scalar product 5-? is maximized. However, if ashould become zero this process is stopped, and it is reversed if ashould be negative. This offers a means of controlling if). a can bedefined as wherein K represents gain. Then,

With this input, relation (10) drives 76-3 to the value C. However, asan applied input, (13) is not necessarily formed altogether external tothe directional computer. Only 5 and C need be introduced externally.Thereafter, the scalar multiplying coeflicient m:K (CfiF) and othersthat modify vectors can be interposed internally.

A horizontal rate of turn in F is asserted by an angular velocity aboutthe vertical direction 5. This is evidentf from FIGURE 5. Such rotationsof '5 are produced directly by the input vector Y, but in opposite senseas noted by relation (5). It is only necessary to assert Y as in orderto give the horizontal projection of ii any desired rate of rotation 9about 5.

The inputs Z and T as the statements (13) and (14) are mutuallycompatible and can enter the directional computer simultaneously insuperposition. They are so applied in FIGURE 3. Alternatively, since Ex?is equivalent to an input Y, (13) and (14) combine as the compositestatement:

The quantities C and Q are readily generated from instrumentationalmeasurements as range, altitude, elevation, bearing, heading, etc. inorder to suit specific automatic guidance requirements. In thisconnection, (2 can be further specialized in (14) or (15) as Q=K FE E(16) where K; is a gain factor and the vector '(7 is supplied by bearingand heading data. (16) becomes zero when '5, 5, and E are coplanar, andit acts to produce this condition. Consequently, 5 will be constrainedto lie in the vertical plane defined by "j and H.

Altogether, the directional computer, by the choice of input vector (asY or Z or both) and the options of modifying it by a few appropriatescalar coeflicients, is able to exert a wide range of directionalcontrols responsive to many data sources in a number of ways. Indifferent problem settings, the same practices apply with differentvector quantities. Of course, Y or E can also be formed as elaboratelyas necessary to begin with.

The directional computer can be realized in conjunction with a referenceframe formed by a physically defined set of three mutually orthogonaldirections, pro viding the component angular velocities (relative tospace) of this reference frame are known by measure ment or otherwise.Relation (l) is stated for such a reference frame F as where thesubscript F on the integral sign identifies F as the reference frame forthe integration (the reference frame relative to which the integrationproceeds) and 0: represents the angular velocity of F. The process ofintegrating relative to a reference frame F means only that thecomponents of the integrand as observed in F are the quantities to beintegrated. The resulting integral outputs have the same reference axesas their respective integrands and express the integral output vector.When the reference frame F is held fixed in space by maintaining itsangular velocity 3 at zero, relation (17) obtains its functionallyequivalent form of relation (I). The integral sign, without subscript,in relation (1) is understood to represent integration relative tospace.

If the vector Y is of the form Y= Z E relation (9) may be applied sothat (17) becomes:

The reference frame may be defined in conjunction with means ofascertaining its angular velocity. As an example, a vertical gyro and agyro compass, as conventionally used in aircraft, together define areference frame having substantially zero components of angularvelocity. The two gyro spin axes define a plane which locates mutualperpendiculars from the spin axis of the vertical gyro both in and outof the plane. The vertical spin axis and its two mutual perpendicularsconstitute a stable, physically defined frame of reference. Anotherreference frame which offers a self-contained capability of expressingits component angular velocities is represented in three rate gyros, thethree gyros being rigidly interrnounted so that their sensitive axes aremutually orthogonal. The three gyros, as a unit, can be mounted ororiented arbitrarily in any manner desired.

The arbitrarily oriented gyro reference frame is analytically the mostgeneral case for instrumentational development. It carries thepossibility of being constrained to zero values of component angularvelocity, either in whole or in part, and also carries the possibilityof being controlled into special values of angular velocity.

The gyro reference directions can be identified as 1, 2, and 3. Thedirectional output 73 is produced as three components [7,, b and breferred to these directions. The angular velocity outputs m m and fromthe three gyros naturally refer to these directions. The vector productX1 or 'B'XCZX'U) must also be expressed in the directions 1, 2, 3. If Yor Z is initially received in an auxiliary frame of reference, acoordinate transformation can be applied to reproduce it by itscomponents Y Y Y or Z Z Z, in the directions 1, 2, 3. Thereupon, 73X? or5X (2X5) can be formed in immediate reference to these directions.Alternatively the output 3 can be transformed into the auxiliaryreference frame of T or Z there to form 5X7 or 5X (7X5). An inversetransformation can then reproduce the vector product by its componentsin the directions 1, 2, 3.

When the input vector Y is expressed in the directions 1, 2, 3, thevector relationship (17) is constructed directly under its componentrelationships:

whe re:

If the vector Y is of the form T=Z F, relation (17) can be convertedinto relation (18), which calls for the following substitutions in (19):

where:

1- 1-l- 2 :i-|- a a When Y or E is received in an auxiliary frame ofreference, the option of transforming the output 5 into the auxiliaryreference directions can be exercised for the purpose of performing thevector product computation. As a typical situation, one of the referenceframes may be supported on one or more pivotal axes relative to theother. FIGURE 1d illustrates such a mount in three degrees of freedom.The auxiliary set of directions is labeled 1', 2, and 3. Y Y Y or Z Z Zrepresent the components of Y or Z in these directions. The vector 5transforms into its components b b b referring to 1', 2', 3' under aseries of resolutions performed on b b and [2 as:

(23) is a standard transformation scheme. Each of the resolutions in(23) is itself a coordinate transformation, the transformation passingbetween adjacent frames of reference having a common pivot axis. P, Q,and R represent intermediate component outputs; 9, and \,'1 representthe angles (FIGURE 1d) that sequentially displace the several referenceframes defined by the mounting and pivotal geometry.

Subsequently, the vector product EXT or F (Z F) can be formed in theframe 1, 2, 3 as zr- W 1' 1''i" 2' 2"i- :i' 3 The vector productcomponents X X X develop into an equivalent vector statement as X X X inthe directions 1, 2, 3 under a transformation inverse to (23) appliedas:

S=X sin z//-X cos l T=X cos L-I-X sin 1/ U=X cos 0-T sin 6 X =X sin 0+Tcos 8 X =U sin S cos 4: X =U cos +S sin 4: (26) With X X and X soproduced, the relations (19) are said by:

b f[ -B)b b b +X 11:

3- b2 3 "J12 W2 2 a t (27) Relations (19) can be satisfied in otherWays. The

Then the relations (19) become:

b,=b= -B)bdz (29) s+ s 0: z-lz and the output '6 belongs to the 1 axisalone. Under this reduction, relation (29) can be dispensed with. Itsfunction is to fix the magnitude 1). But, in being a single component,the magnitude 1) can be assigned any constant value at once. Moreover,relations (30) and (31) are indifferent to this constant. The constantis necessary only in a process of transforming the vector into anotherreference frame and in scaling Y and Y to express dE/dt. In the absenceof an expressed constant, Y and Y express dF/dz with the implicationthat b is unity.

Relations (30) and (31) assert a requirement for specific control overthe angular velocity components of the reference frame about the twoaxes perpendicular to the reference axis expressing 5. Hence, thisreference frame 1, 2, 3 is serving as a specialized computing element,and the input vector Y invariably originates in a separate and distinctframe of reference. The components Y and Y have to be developed under acoordinate transformation. The coordinate transformation is subsequentlyaffected by its own component outputs and is a functional participant inthis process of establishing relations (19).

FIGURE 2 illustrates the situation. The vector passes to the gyroreference frame 1, 2, 3 from the supporting frame of reference 1', 2',3'. The transformation proceeds as:

Thereafter, the gyro reference frame is driven with the angular rates Yand -Y about its 2 and 3 axes. This in turn affects the angles 0 and Q5.

The input vector T is asserted as 2X5 in relations (30) and (31) whenthe substitutions (21) are applied with recognition of the conditions(28), so that:

But A/B is just b and b is an arbitrary scale constant. In etfect, thesubstitutions (21) enter relations (30) and (31) as:

The components Z and Z come into the gyro reference frame through acoordinate transformation of the input vector E that is just like thetransformation of Y.

The figures present some varied embodiments of the directional computer.FIGURE 1 and any one of FIG- URES 1a, 1b, and 1c illustrate examplesfrom relations 8 (19) to (22). FIGURES 1 and 1d illustrate the exampleof relations (23) to (27). FIGURE 2 illustrates the example of relations(30), (31), and (32) and the like example of relations (32) and (34).FIGURE 6 illustrates a manner of forming the input vector (15) forincorporation in FIGURE 2.

The electro-mechanical integrators in FIGURE 1 produce their rotationaloutputs as signals b b and b These integral outputs each position anumber of linear potentiometers to yield products of multiplication. Thefirst two potentiometers (from right to left, FIGURE 1) on each shaftare series connected through feedback amlifiers and are energized by acommon transformer output e to give the signals eb 219 and eb as theirend outputs. These three outputs all feed back, together with areference voltage input E, into the amplifier that produces e. Throughthe high gain of the amplifier, the finite output e is produced underthe condition:

L" E "mwwwfw and the first stage outputs from the series connectedpotentiometers are signals proportional to b /b b /b and 17 /17 Thesesignals are delivered to the amplifier input stage of their respectiveintegrators in accord with relations (19). The same applies to thesignals b -b and b;, that are tapped from the remaining threepotentiometers of FIGURE 1. The signals scale to the desiredcoefficients A and B of relations (19) through the values to inputsumming resistors contained in the amplifiers. The further electricalsignals that are delivered to the integrators are identified by thenumerals 1 to 12 shown both in FIGURE 1 and in any one of the FIGURES1a, lb, 1c, and Id.

In FIGURE In, six more potentiometers are positioned by the three shaftrotations b b and h The potentiometers are energized by the signals 1(01 Y i (m -l- Y and *:(w +Y that are so combined in feedbackamplifiers. The six products of multiplication return as inputs to theintegrators of FIGURE 1 in accordance with relations (19). Only six ofthe twelve electrical return lines are utilized.

FIGURE 1b is functionally the equivalent of FIGURE 10. Twelve ratherthan six potentiometer multiplications by b b and b are generated underseparate energizing by iw iwg, iw and iY iY iY The products are allproperly fed back and summed at the integrators of FIGURE 1. Theseparate development of the products in b and Y allows the components ofEXT to be formed to give dW/dt.

FIGURE 10 repeats FIGURE 1b in its manner of producing the six productsin G1 and b and applying them at the integrators of FIGURE 1.Thereafter, FIG- URE 1c develops the substitutions asserted by relations21) and (22). The signals Z Z Z successively energize threepotentiometers on the shaft outputs b b and h The three products formedare submitted to a feedback amplifier for summation as the output Thisoutput is applied to three more potentiometers to generate the products(F-'Z)b ('5-Z)b and (F Z)b These product signals and the signals Z Z andZ enter the integrators of FIGURE 1 in accord With relations (21) and(19).

In FIGURE III, the vector product P7X? or 3X (7X71) is developed in anauxiliary frame of reference directions 1', 2', 3' wherein the vector Yor /1 is expressed by the signals Y Y Y or Z Z Z The signals b 9 b and bcome from FIGURE 1. Three resolvers, each capable of performing thecomputations:

:(x sin a-l-y cos a) a=any angle :(x cos a-y sin a) on inputs x and y inany desired pairing of and/or signs, apply the transformation (23) andreproduce the output direction 3 as component signals 1);, b b referredto the auxiliary reference frame. Three positioning servos cansubsequently reproduce the signals b b and b as equivalent shaftrotations and the vector product computations (24) can proceed in themanner represented in FIGURES 1b and 1c. The signal summations noted asX X X in relations (24) form at the input terminals of three moreresolvers performing the inverse transformation (26). The vector productis thus produced as signals X X X referring to the gyro referencedirections 1, 2, 3. X X and X enter the integrators of FIGURE 1 inaccordance with relations (27). The cross product signals 01 and b alsoenter the intcgrators of FIGURE 1 in accordance with relations (27),their formation being processed in the manner represented in FIGURES lband 10.

In any of the foregoing embodiments, the product formations in m m and bb b; are eliminated in Whole or in part by the special circumstance thatthe reference frame is stabilized so that one or more of its componentangular velocities is zero. If, also, by the nature of the input vectorin the particular reference frame utilized, one or two of the inputvector components are consistently zero, all the computations regardingthem in that reference frame go out.

FIGURE 2 shows an embodiment applying a controllable reference frame asfurnished by a free gyro. The gyro spin axis, the inner gimbal axis, andtheir mutually perpendicular direction provide the reference directions1, 2, and 3 respectively, the spin axis 1 being the output direction.The structure mounting the gyro defines the auxiliary referencedirections 1', 2', 3. Two resolvers receiving the input vector T ascomponents Y Y Y in the auxiliary frame of reference perform thetransformation (32) to reproduce T by its components Y Y Y; in the gyroreference frame. The component Y is superfluous, however. The necessarycomponents are Y and Y Y and Y must satisfy relations (30) and (31) bydriving the gyro reference frame into component angular velocities thatare in equal and opposite proportions to themselves. For this reason, Yis first diminished through a resolver by the multiplying factor cos 0.Then Y cos 0 and Y are amplified to precise power levels and energizelinear torqucmotors acting on the gyro about the 3 and 2 axes. Y drivingthrough the 2 axis, rotates the gyro about the 3 axis at a directlyproportional rate. Y cos 0, driving through 3, is offset by angle 0 fromthe gyros 3 axis. While causing the gyro to rotate about the 2 axis, Ycos 9 induces a reaction torque normal to the pivotal axes of the outergimbal. This reaction torque and the applied torque about 3' give theirresultant about the 3 axis and act together to rotate the gyro. Theresultant exceeds the applied torque by the factor sec 9. But Y has beenreciprocally attenuated so that the gyro rotates about its 2 axis at arate directly proportional [0 Y2.

The relations (34) are produced instead of (30) and (3l) in FIGURE 2when Y and Y; are identified as Z and Z and their roles in rotating thegyro reference frame are interchanged. In either case, the rotationalrates must have the functionally correct sense and the torquemotor inputconnections must observe proper electrical polarities.

FIGURE 6 illustrates the facility with which the directional computeraccommodates various, purposeful 10 input vectors. In this example, theinput vector of FIG- URE 2 is developed as the quantity (15).

By FIGURE 6, the two left-most resolvers in FIG- URE 2 are consigned toa transformation between a vertical gyro and the free gyro. Thistransformation refers a signal g to a physically demonstrated verticaldirection and resolves it in stages into the free gyro frame ofreference. Thereafter, the component outputs g g and g are submitted toadditional computations that modify them into components conforming tothe vector quantity (15). These components are developed for the 2 and 3axes and drive the free gyro in just the manner illustrated with Y and Yin FIGURE 2.

The upper resolver in FIGURE 6 is rotated under the feedback of one ofits outputs and acquires the angle '4 that produces its other output asthe resultant of the inputs g and g The same angle 1 positions the lowerresolver. The indicated input and output relationships for the lowerresolver are self-evident under the relationships and 92 COS 71 Q Theinventor claims:

1. A directional computer comprising a physically defined referenceframe, means of obtaining a plurality of input signals in representationof any desired values, said input signals being referred to saidreference frame to express an input vector, means of generating aplurality of signals describing the angular motion of said referenceframe in space, said signals of angular motion indicating the angularvelocity vector of said reference frame, and means to receive the saidinput and angular motion signals and generate a plurality of outputsignals expressing an outer vector in relation to said ref erence frameas the integral of the cross product vectors between the said outputvector and the said angular velocity vector and between the said outputvector and the said input vector, whereby the said output vector rotatesaround the said input vector.

2. A directional computer comprising a physically defined, stabilizedreference frame having means of sensing and suppressing its angularvelocity, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said reference frame to express an input vector, and means to receivethe said input signals and generate a plurality of output signalsexpressing an output vector in relation to said reference frame as theintegral of the cross product vector between the said output vector andthe said input vector, whereby the said output vector rotates around thesaid input vector.

3. A directional computer comprising a physically defined, stabilizedreference frame having means of sensing and suppressing its angularvelocity, the axes of said reference frame being denoted by the numerals1, 2, 3, means of obtaining an input signal in representation of anydesired value, said input signal being referred to said axis 3 toexpress an input vector, and means to receive the said input signal andgenerate a pair of output signals expressing an output vector inreference to the said axes 1 and 2 as the integral of the cross productvector between the said output vector and the said said output vector isdriven by the said input signal to rotate around the said axis 3.

4. A directional computer comprising a physically defined firstreference frame and a second reference frame defined with respect to thesaid first frame by means to perform a coordinate transformationtherebetween, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said second reference frame to express an input vector, means ofgenerating a plurality of signals describing the angular motion of saidfirst reference frame in space, said signals of angular motion indicat*ing the angular velocity vector of said first reference frame, andmeans, including said coordinate transformation means, to receive thesaid input and angular motion signals and generate a plurality of outputsignals expressing an output vector in relation to said first referenceframe as the integral of the cross product vectors between the saidoutput vector and the said angular velocity vector and between the saidoutput vector and the said input vector, whereby the said output vectorrotates around the said input vector.

5. A directional computer comprising a physically defined firstreference frame and a second reference frame defined with respect to thesaid first frame by means to perform a coordinate transformationtherebetween, means of obtaining an input signal in representation ofany desired value, said input signal being referred to an axis of saidsecond reference frame to express an input vector, means of generating aplurality of signals describing the angular motion of said firstreference frame in space, said signals of angular motion indicating theangular velocity vector of said first reference frame, and means,including said coordinate transformation means, to receive the saidinput and angular motion signals and generate a plurality of outputsignals expressing an output vector in relation to said first referenceframe as the integral of the cross product vectors between the saidoutput vector and the said angular velocity vector and between the saidoutput vector and the said input vector, whereby the said output vectorrotates around the said input vector.

6. A directional computer comprising a physically defined, stabilizedreference frame having means of sensing and suppressing its angularvelocity, a second reference frame defined with respect to the saidstabilized frame by means to perform a coordinate transformationtherebetween, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said second reference frame to express an input vector, and means,including said coordinate transformation means, to receive the saidinput signals and generate a plurality of output signals expressing anoutput vector in relation to said stabilized reference frame as theintegral of the cross product vector between the said output vector andthe said input vector, whereby the said output vector rotates around thesaid input vector.

7. A directional computer comprising a physically defined, stabilizedreference frame having means of sensing and suppressing its angularvelocity, a second reference frame defined with respect to the saidstabilized frame by means to perform a coordinate transformationtherebetween, means of obtaining an input signal in representation ofany desired value, said input signal being referred to an axis of saidsecond reference frame to express an input vector, and means, includingsaid coordinate transformation means, to receive the said input signaland generate a plurality of output signals expressing an output vectorin relation to said stabilized reference frame as the integral of thecross product vector between the said output vector and the said inputvector, whereby the said output vector rotates around the said inputvector.

8. A directional computer comprising a physically deinput vector,whereby the fined reference frame, mens of obtaining a plurality ofinput signals in representation of any desired values, said inputsignals being referred to said reference frame to express an inputvector, means of generating a plurality of signals describing theangular motion of said reference frame in space, said signals of angularmotion indicating the angular velocity vector of said reference frame,and means to receive the said input and angular motion signals andgenerate a plurality of output signals expressing an output vector inrelation to said reference frame as the integral of a vector quantityconsisting of the said input vector, the negative value of the saidoutput vector multiplied by the scalar product between the said inputand output vectors, and the cross product vector between the said outputvector and the said angular velocity vector, whereby the said outputvector rotates around a direction perpendicular both to the said outputvector and the said input vector.

9. A directional computer comprising a physically defined, stabilizedrefercnce frame, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said reference frame to express an input vector, and means to receivethe said input signals and generate a plurality of output signalsexpressing an output vector in relation to said reference frame as theintegral of a vector quantity consisting of the said input vector andthe negative value of the said output vector multiplied by the scalarproduct between the said input and output vectors, whereby the saidoutput vector rotates around a direction perpendicular both to the saidoutput vector and the said input vector.

10. A directional computer comprising a physically defined, stabilizedreference frame, means of obtaining an input signal in representation ofany desired value, said input signal being referred to an axis of saidreference frame to express an input vector, and means to receive thesaid input signal and generate a plurality of output signals expressingan output vector in relation to said reference frame as the integral ofa vector quantity consisting of the said input vector and the negativevalue of the said output vector multiplied by the scalar product betweenthe said input and output vectors, whereby the said output vectorrotates around a direction perpendicular both to the said output vectorand the said input vector.

11. A directional computer comprising a physically defined firstreference frame and a second reference frame defined with respect to thesaid first frame by means to perform a coordinate transformationtherebetween, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said second reference frame to express an input vector, means ofgenerating a plurality of signals describing the angular motion of saidfirst reference frame in space, said signals of angular motionindicating the angular velocity vector of said first reference frame,and means, including said coordinate transformation means, to receivethe said input and angular motion signals and generate a plurality ofoutput signals expressing an output vector in relation to said firstreference frame as the integral of a vector quantity consisting of thesaid input vector, the negative value of the said output vectormultiplied by the scalar product between the said input and outputvectors, and the cross product vector between the said output vector andthe said angular velocity vector, whereby the said output vector rotatesaround a direction perpendicular both to the said output vector and thesaid input vector.

12. A directional computer comprising a physically defined firstreference frame and a second reference frame defined with respect to thesaid first frame by means to perform a coordinate transformationtherebetween, means of obtaining an input signal in representation ofany desired value, said input signal being referred to an axis of saidsecond reference frame to express an input vector,

means of generating a plurality of signals describing the angular motionof said first reference frame in space, said signals of angular motionindicating the angular velocity vector of said first reference frame,and means, including said coordinate transformation means, to receivethe said input and angular motion signals and generate a plurality ofoutput signals expressing an output vector in relation to said firstreference frame as the integral of a vector quantity consisting of thesaid input vector, the negative value of the said output vectormultiplied by the scaler product between the said input and outputvectors, and the cross product vector between the said output vector andthe said angular velocity vector, whereby the said output vector rotatesaround a direction perpendicular both to the said output vector and thesaid input vector.

13. A directional computer comprising a physically defined, stabilizedreference frame and a second reference frame defined with respect to thesaid stabilized frame by means to perform a coordinate transformationtherebetween, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said second reference frame to express an input vector, and means,including said coordinate transformation means, to receive the saidinput signals and generate a plurality of output signals expressing anoutput vector in relation to said stabilized reference frame as theintegral of a vector quantity consisting of the said input vector andthe negative value of the said output vector multiplied by the scalarproduct between the said input and output vectors, whereby the saidoutput vector rotates around a direction perpendicular both to the saidoutput vector and the said input vector.

14. A directional computer comprising a physically defined, stabilizedreference frame and a second reference frame defined with respect to thesaid stabilized frame by means to perform a coordinate transformationtherebetween, means of obtaining an input signal in representation ofany desired value, said input signal being referred to an axis of saidsecond reference frame to express an input vector, and means, includingsaid coordinate transformation means, to receive the said input signaland generate a plurality of output signals expressing an output vectorin relation to said stabilized reference frame as the integral of avector quantity consisting of the said input vector and the negativevalue of the said output vector multiplied by the scalar product betweenthe said input and output vectors, whereby the said output vectorrotates around a direction perpendicular both to the said output vectorand the said input vector.

15. A directional computer comprising a physically defined, stabilizedreference frame and a separate reference axis defined with respect tothe said stabilized frame by means to perform coordinate resolutionstherebetween, means of obtaining an input signal in representation ofany desired value, said input signal being referred to said separateaxis to express an input vector, and means, including said coordinateresolution means, to receive the said input signal and generate aplurality of output signals expressing an output vector in relation tosaid stabilized reference frame as the integral of a vector quantityconsisting of the said input vector and the negative value of the saidoutput vector multiplied by the scalar product between the said inputand output vectors, whereby the said output vector rotates around adirection perpendicular both to the said output vector and the saidinput vector.

16. A directional computer comprising a physically defined referenceframe and a separate reference axis defined with respect to the saidreference frame by means to perform coordinate resolutions therebetween,means of obtaining an input signal in representation of any desiredvalue, said input signal being referred to said separate axis to expressan input vector, means of generating a plurality of signals describingthe angular motion of said reference frame in space, said signals ofangular motion indicating the angular velocity vector of said referenceframe, and means, including said coordinate resolution means, to receivethe said input and angular motion signals and generate a plurality ofoutput signals expressing an output vector in relation to said referenceframe as the integral of a vector quantity consisting of the said inputvector, the negative value of the said output vector multiplied by thescalar product between the said input and output vectors, and the crossproduct vector between the said output vector and the said angularvelocity vector, whereby the said output vector rotates around adirection perpendicular both to the said output vector and the saidinput vector.

17. A directional computer comprising means of generating a plurality ofsignals describing the angular motion of a physical reference frame inspace, said reference frame being defined by the said means, saidsignals of angular motion indicating the angular velocity vector of saidreference frame, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said reference frame to express an input vector, and means to receivethe said input and angular motion signals and generate a plurality ofoutput signals expressing an output vector in relation to said referenceframe as the integral of the cross product vectors between the saidoutput vector and the said angular velocity vector and between the saidoutput vector and the said input vector, whereby the said output vectorrotates around the said input vector.

18. A directional computer comprising means of generating a plurality ofsignals describing the angular motion of a physical reference frame inspace, said reference frame being defined by the said means, saidsignals of angular motion indicating the angular velocity vector of saidreference frame, a second reference frame defined with respect to thesaid physical frame by means to perform a coordinate transformationtherebetween, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said second reference frame to express an input vector, and means,including said coordinate transformation means, to receive the saidinput and angular motion signals and generate a plurality of outputsignals expressing an output vector in relation to said physicalreference frame as the integral of the cross product vectors between thesaid output vector and the said angular velocity vector and between thesaid output vector and the said input vector, whereby the said outputvector rotates around the said input vector.

19. A directional computer comprising, means of generating a pluralityof signals describing the angular motion of a physical reference framein space, said reference frame being defined by the said means, saidsignals of angular motion indicating the angular velocity vector of saidreference frame, a second reference frame defined with respect to thesaid physical frame by means to perform a coordinate transformationtherebetween, means of obtaining an input signal in representation ofany desired value, said input signal being referred to an axis of saidsecond reference frame to express an input vector, and means, includingsaid coordinate transformation means, to receive the said input andangular motion signals and generate a plurality of output signalsexpressing an output vector in relation to said physical reference frameas the integral of the cross product vectors be tween the said outputvector and the said angular velocity vector and between the said outputvector and the said input vector, whereby the said output vector rotatesaround the said input vector.

20. A directional computer comprising means of generating a plurality ofsignals describing the angular motion of a physical reference frame inspace, said reference frame being defined by the said means, saidsignals of angular motion indicating the angular velocity vector of saidreference frame, means of obtaining a plurality of input signals inrepresentation of. any desired values, said input signals being referredto said reference frame to express an input vector, and means to receivethe said input and angular motion signals and generate a plurality ofoutput signals expressing an output vector in relation to said referenceframe as the integral of a vector quantity consisting of the said inputvector, the negative value of the said output vector multiplied by thescalar product between the said input and output vectors, and the crossproduct vector between the said output vector and the said angularvelocity vector, whereby the said output vector rotates around adirection perpendicular both to the said output vector and the saidinput vector.

21. A directional computer comprising means of generating a plurality ofsignals describing the angular motion of a physical reference frame inspace, said reference frame being defined by the said means, saidsignals of angular motion indicating the angular velocity vector of saidreference frame, a second reference frame defined with respect to thesaid physical frame by means to perform a coordinate transformationtherebetween, means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said second reference frame to express an input vector, and means,including said coordinate transformation means, to receive the saidinput and angular motion signals and generate a plurality of outputsignals expressing an output vector in relation to said physicalreference frame as the integral of a vector quantity consisting of thesaid input vector, the negative value of the said output vectormultiplied by the scalar product between the said input and outputvectors, and the cross product vector between the said output vector andthe said angular velocity vector, whereby the said output vector rotatesaround a direction perpendicular both to the said output vector and thesaid input vector.

22. A directional computer comprising means of generating a plurality ofsignals describing the angular motion of a physical reference frame inspace, said reference frame being defined by the said means, saidsignals of angular motion indicating the angular velocity vector of saidreference frame, a second reference frame defined with respect to thesaid physical reference frame by means to perform a coordinatetransformation therebetween, means of obtaining an input signal inrepresentation of any desired value, said input signal being referred toan axis of said second reference frame to express an input vector, andmeans, including said coordinate transformation means, to receive thesaid input and angular motion signals and generate a plurality of outputsignals expressing an output vector in relation to said physicalreference frame as the integral of a vector quantity consisting of thesaid input vector, the negative value of the said output vectormultiplied by the scalar product between the said input and outputvectors, and the cross product vector between the said output vector andthe said angular velocity vector, whereby the said output vector rotatesaround a direction perpendicular both to the said output vector and thesaid input vector.

23. A directional computer comprising means of generating a plurality ofsignals describing the angular motion of a physical reference frame inspace, said reference frame being defined by the said means, saidsignals of angular motion indicating the angular velocity vector of saidreference frame, a separate reference axis defined with respect to thesaid reference frame by means to perform coordinate resolutionstherebetween, means of obtaining an input signal in representation ofany desired value, said input signal being referred to said separateaxis to express an input vector, and means, including said coordinateresolution means, to receive the said input and angular motion signalsand generate a plurality of output signals expressing an output vectorin relation to said reference frame as the integral of a vector quantityconsisting of the said input vector, the negative value of the saidoutput vector multiplied by the scalar product between the said inputand output vectors, and the cross product vector between the said outputvector and the said angular velocity vector, whereby the said outputvector rotates around a direction perpendicular both to the said outputvector and the said input vector.

24. A directional computer comprising a physically defined referenceframe adapted to rotate about two of its axes, means of receivingangular rate signals and generating corresponding rotations of saidreference frame relative to space about said rotational axes, meansresponsive to said rotations to produce coordinate resolutions betweenthe said rotational axes and the axes of a second reference frame,whereby said second reference frame is defined with respect to the saidrotational axes, and means of obtaining a plurality of input signals inrepresentation of any desired values, said input signals being referredto said sec-0nd reference frame to express an input vector, the saidcoordinate resolutions being applied to the said input signals toproduce transformed signals expressing the two components of the saidinput vector relating to the said rotational axes, the said transformedsignals forming the said angular rate signals, whereby the directionperpendicular to the said rotational axes in said physically definedreference frame is caused to rotate in a known way with respect to thesaid input vector,

25. A directional computer comprising a physically defined referenceframe adapted to rotate about two of its axes, means of receivingangular rate signals and generating corresponding rotations of saidreference frame relative to space about said rotational axes, meansresponsive to said rotations to produce coordinate resolutions betweenthe said rotational axes and a separate reference axis, whereby saidseparate axis is defined with respect to the said rotational axes, andmeans of obtaining an input signal in representation of any desiredvalue, said input signal being referred to said separate axis to expressan input vector, the said coordinate resolutions being applied to thesaid input signal to produce transformed signals expressing the twocomponents of the said input vector relating to the said rotationalaxes, the said transformed signals forming the said angular ratesignals, whereby the direction perpendicular to the said rotational axesin said physically defined reference frame is caused to rotate in aknown way with respect to the said input vector.

References Cited by the Examiner UNITED STATES PATENTS MALCOLM A.MORRISON, Primary Examiner.

I. KESCHNER, Assistant Examiner.

UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No '3 ,319,052 May 9 1967 George Arshal I It is certified that error appears inthe above identified patent and that said Letters Patent are herebycorrected as shown below: Column 2, line 67, b Z b" should read b (Z b)Column 3, line 1, "variable" should read valuable Column 5, line 61, "Yshould read Y Column 7, line 1, "w b should read w b Column 9 line 22after signals insert in Column 10, line 22, "n" should read sin n line47, "outer" should read output Column 12, line 1, "mens" should readmeans Column 13, line 10, "scaler should read scalar Signed and sealedthis 24th day of March 1970.

(SEAL) Attest: Edward M. Fletcher, Jr. WILLIAM E. SCHUYLER, JR.

Attesting Officer Commissioner of Patents

2. A DIRECTIONAL COMPUTER COMPRISING A PHYSICALLY DEFINED, STABILIZEDREFERENCE FRAME HAVING MEANS OF SENSING AND SUPPRESSING ITS ANGULARVELOCITY, MEANS OF OBTAINING A PLURALITY OF INPUT SIGNALS INREPRESENTATION OF ANY DESIRED VALUES, SAID INPUT SIGNALS BEING REFERREDTO SAID REFERENCE FRAME TO EXPRESS AN INPUT VECTOR, AND MEANS TO RECEIVETHE SAID INPUT SIGNALS AND GENERATE A PLURALITY OF OUTPUT SIGNALSEXPRESSING AN OUTPUT VECTOR IN RELATION TO SAID REFERENCE FRAME AS THEINTEGRAL OF THE CROSS PRODUCT VECTOR BETWEEN THE SAID OUTPUT VECTOR ANDTHE SAID INPUT VECTOR, WHEREBY THE SAID OUTPUT VECTOR ROTATES AROUND THESAID INPUT VECTOR.